# Is this a linear equation 5x+6y=3x-2?

Yes, it is a linear equation in two variables $x$ & $y$.

#### Explanation:

The generalized linear equation in two variables $x$ & $y$ is given as

$a x + b y + c = 0$

The above equation shows a straight line on $X Y$ plane.

The given equation

$5 x + 6 y = 3 x - 2$

or $2 x + 6 y + 2 = 0$

or $x + 3 y + 1 = 0$

The above equation is in form of linear equation: $a x + b y + c = 0$

Jul 28, 2018

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#### Explanation:

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Linear Equations are referred to as the equations of degree one,

where the variable's exponent ( or Power) is the value one.

This is the most common form of a linear equation.

It takes the form:

color(red)(y=mx+b, where

color(red)(m is the Slope or the Gradient

color(red)(b is the y-intercept

General Form:

IF there are two variables in a linear equation it takes the form:

color(red)(Ax+By=C

To find the x-intercept, set color(blue)(y=0

To find the y-intercept, set color(blue)(x=0

color(blue)("Note :"

For both of the above forms, we can construct a data table and create a graph.

Now, we will get back to the problem given to us:

color(red)(5x+6y = 3x-2

This is a linear equation with two variables

We can combine like terms and simplify the equation as
follows:

$\Rightarrow 5 x + 6 y = 3 x - 2$

Subtract color(red)(3x from both sides of the equation to balance the equation.

$\Rightarrow 5 x + 6 y - 3 x = 3 x - 2 - 3 x$

$\Rightarrow 5 x + 6 y - 3 x = \cancel{3 x} - 2 - \cancel{3 x}$

color(blue)(rArr 2x+6y=(-2)

Hence, we can conclude that the equation

color(red)(5x+6y = 3x-2

is indeed a linear equation.

y-intercept can be found by setting color(red)(x=0

$\Rightarrow 2 x + 6 y = \left(- 2\right)$

$\Rightarrow 2 \left(0\right) + 6 y = \left(- 2\right)$

$\Rightarrow 6 y = - 2$

Divide both sides by color(red)(6

$\Rightarrow \frac{6 y}{6} = \frac{- 2}{6}$

$\Rightarrow y = - \frac{1}{3}$

Hence we understand that color(blue)((0,-1/3) is the y-intercept

To find the x-intercept, set color(red)(y=0

$\Rightarrow 2 x + 6 y = \left(- 2\right)$

$\Rightarrow 2 x + 6 \left(0\right) = \left(- 2\right)$

$\Rightarrow 2 x = - 2$

Divide both sides of the equation by color(red)(2

$\Rightarrow \frac{2 x}{2} = \frac{- 2}{2}$

$\Rightarrow \frac{\cancel{2} x}{\cancel{2}} = \frac{- 2}{2}$

$\Rightarrow x = - 1$

Hence we understand that color(blue)((-1,0) is the x-intercept

We can verify these results using a graph as given below:

I hope this explanation is helpful.